Transition to turbulence and unstable nonlinear magnetohydrodynamic flow states in ducts with a transverse magnetic field

在各种流动系统中，向湍流的过渡是很重要的。当雷诺数Re增加时，它通常发生在有壁面的流动中，即不考虑层流状态的不稳定性。这种转变通常用动力系统理论的框架来解释。流态被看作是可能流态相空间中的一个点。层流和湍流有不同的吸引盆地，它们被一个超表面（混沌边缘）分开。在这个表面上，流动既不向层流状态发展，也不向湍流状态发展，而是向表面内的一个吸引子发展，这个吸引子被称为边缘状态。在向湍流过渡的过程中，流动可能首先到达边缘状态附近，然后最终成为湍流。边缘状态对应于非平凡的时变流，它比湍流解具有更简单的动力学。因此，它们可以作为研究过渡物理机制的模型。例如，边缘状态可以对应于类波解、时间周期解或混沌解。它们在许多原型剪切流中进行了详细的研究，但很少考虑磁流体动力学（MHD）流动，尽管这些流动中的转捩也是亚临界的。在MHD流动中，除了雷诺数作为速度的无量纲参数外，哈特曼数Ha作为磁场的第二个参数出现。在该项目中，将确定MHD管道流动的湍流路径和过渡机制，这在很大程度上是未知的。所要研究的具体形态是具有矩形截面和横向均匀磁场的管道中的流动。这些流动是三维的，其特征是壁面上平行和垂直于磁场的边界层，分别被称为shercliff层和Hartmann层。流动将通过高分辨率直接数值模拟（DNS）和稳定性分析进行研究。首先，在re - ha -A参数空间中确定层流和湍流状态的边界，其中A为初始状态的扰动幅度。在第二步中，边缘状态将由DNS确定，并分析其动态和特定属性。在第三步中，将研究Re和Ha变化时流动状态的分岔，从而产生边缘状态。由此，人们也可以更好地理解为什么过渡是由一个基于哈特曼层厚度的参数决定的，尽管湍流首先出现在Shercliff层中。我们的研究结果对冶金和导热流体的电磁流动控制具有重要意义。

Transition to turbulence is important in various flow systems. Upon increasing the Reynolds number Re, it typically occurs sub-critically in wall-bounded flows, i.e. irrespective of an instability of the laminar flow state. The transition is typically interpreted in the framework of dynamical systems theory. A flow state is regarded as a point in the phase space of possible flow states. Laminar and turbulent flow have different basins of attraction that are separated by a hypersurface (edge of chaos). On this surface the flow develops neither towards the laminar nor towards the turbulent state, but to an attractor within the surface that is called the edge state. During transition to turbulence, the flow may first reach the vicinity of the edge state before it eventually becomes turbulent. Edge states correspond to non-trivial time-dependent flows that have simpler dynamics than turbulent solutions. They can therefore serve as models for studyingthe physical mechanisms of transition. Edge states can, e.g., correspond to wave-like, time-periodic or chaotic solutions. They were studied in detail for a number of prototypical shear flows, but have hardly ever been considered for magnetohydrodynamic (MHD) flows although transition in these flows is also sub-critical. In MHD flows, the Hartmann number Ha appears as a second parameter for the magnetic field in addition to the Reynolds number as a dimensionless parameter for the velocity. In the project, the route to turbulence and the mechanisms of transition in MHD duct flows will be determined that are largely unknown. The specific configurations to be studied are flows in ducts with rectangular cross-section and transversal homogeneous magnetic field. These flows are three-dimensional and characterized by boundary layers on the walls parallel and perpendicular to the magnetic field that are calledShercliff and Hartmann layers, respectively. The flows will be investigated by highly resolved direct numerical simulations (DNS) and stability analysis. First, the boundary between laminar and turbulent states will be determined in the Re-Ha-A-parameter space, where A is theperturbation amplitude of the initial state. In the second step, edge states will be determined by DNS and their dynamics and specific properties will be analyzed. In the third step, the bifurcations of flow states upon changing of Re and Ha will be investigated that give rise to edge states. By that one can also obtain a better understanding why transition is determined by a parameter that is based on the thickness of the Hartmann layers although turbulence appears first in the Shercliff layers. Our results are relevant for electromagnetic flow control in metallurgical and heat transfer applications with conducting liquids.